INTERNATIONAL STANDARD ISO 11648-1 First edition 2003-03-15 Part 1: General principles Aspects statistiques de l'échantillonnage des matériaux en vrac — Partie 1: Principes généraux `,,`,-`-`,,`,,`,`,,` - Statistical aspects of sampling from bulk materials — Reference number ISO 11648-1:2003(E) Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 Not for Resale ISO 11648-1:2003(E) `,,`,-`-`,,`,,`,`,,` - PDF disclaimer This PDF file may contain embedded typefaces In accordance with Adobe's licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing In downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy The ISO Central Secretariat accepts no liability in this area Adobe is a trademark of Adobe Systems Incorporated Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters were optimized for printing Every care has been taken to ensure that the file is suitable for use by ISO member bodies In the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below © ISO 2003 All rights reserved Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO's member body in the country of the requester ISO copyright office Case postale 56 • CH-1211 Geneva 20 Tel + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyright@iso.org Web www.iso.org Published in Switzerland ii Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) Contents Page Foreword iv Scope Normative references Terms, definitions, symbols and abbreviated terms Purpose and application of statistics in sampling from bulk material 11 Particular problems for sampling bulk materials 11 Differences between particulates, liquids and gases 13 Experimental methods for obtaining variance components at various stages of sampling 14 Adjusting the sampling plan to obtain desired precision 19 Estimating precision 20 10 Checking for bias 20 11 Precision and bias at measurement stage 22 Annex A (informative) Explanatory notes on definitions 23 Annex B (informative) Fully-nested experiments 28 Annex C (informative) Statistical analysis of serial data 36 Annex D (normative) Estimating precision 74 Annex E (normative) Checking for bias 78 Bibliography 91 iii © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale `,,`,-`-`,,`,,`,`,,` - Introduction v ISO 11648-1:2003(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies) The work of preparing International Standards is normally carried out through ISO technical committees Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part The main task of technical committees is to prepare International Standards Draft International Standards adopted by the technical committees are circulated to the member bodies for voting Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights ISO shall not be held responsible for identifying any or all such patent rights ISO 11648-1 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods `,,`,-`-`,,`,,`,`,,` - ISO 11648 consists of the following parts, under the general title Statistical aspects of sampling from bulk materials: Part 1: General principles Part 2: Sampling of particulate materials It is the intention of ISO/TC 69/SC to develop additional parts under this general title for the sampling of liquids and gases, if the need exists iv Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) Introduction This first part of ISO 11648 gives a broad outline of the statistical aspects of sampling from bulk material `,,`,-`-`,,`,,`,`,,` - International Standards dealing with the methods for sampling for bulk materials, such as solid fuels, iron ores, etc., have already been published and some of these are being revised by the responsible technical committees This International Standard provides a source of information for technical terms and sampling techniques for types of bulk materials for which International Standards on sampling have not yet been written This International Standard may also act as a bridge for mutual understanding of terms and methods between Technical Committees v © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale `,,`,-`-`,,`,,`,`,,` - Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale INTERNATIONAL STANDARD ISO 11648-1:2003(E) Statistical aspects of sampling from bulk materials — Part 1: General principles Scope This part of ISO 11648 establishes the general principles for the application and statistical treatment of the sampling of bulk materials It also provides general guidance and examples for estimating necessary variances and checking precision and bias when the average value of a quality characteristic is investigated Furthermore, this part of ISO 11648 gives information relating to the statistical analyses of serial data, by the use of variograms and correlograms This part of ISO 11648 also defines the basic terms with definitions for the sampling of bulk materials These terms are necessary for providing a better understanding of sampling techniques as well as making it easier to fulfil requirements NOTE Part of ISO 11648 is applicable to particulate materials in bulk Normative references The following referenced documents are indispensable for the application of this document For dated references, only the edition cited applies For undated references, the latest edition of the referenced document (including any amendments) applies ISO 565, Test sieves — Metal wire cloth, perforated metal plate and electroformed sheet — Nominal sizes of openings ISO 3534 (all parts), Statistics — Vocabulary and symbols ISO 5725 (all parts), Accuracy (trueness and precision) of measurement methods and results 3.1 Terms, definitions, symbols and abbreviated terms Terms and definitions For the purposes of this document, the terms and definitions given in ISO 3534 and the following apply NOTE The text 〈bulk material〉 shown after terms means the definition given is confined to the field of bulk sampling NOTE For further information on definitions, see Annex A 3.1.1 bulk material amount of material within which component parts are not initially distinguishable on the macroscopic level `,,`,-`-`,,`,,`,`,,` - © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) 3.1.2 sample 〈bulk material〉 subset of a specified population made up of one or more sampling units 3.1.3 sampling act of drawing or constituting a sample 3.1.4 simple random sampling sampling where a sample of n sampling units is taken from a population in such a way that all combinations of n sampling units have the same probability of being taken NOTE In bulk material sampling, if the sampling unit is an increment, the positioning, delimitation and extraction of increments should ensure that all sampling units have an equal probability of being selected 3.1.5 stratum mutually exclusive and exhaustive sub-population considered to be more homogeneous with respect to the characteristics investigated than the total population EXAMPLES In bulk material, strata, based on time, mass and space, are typically production periods (e.g 15 min); production masses (e.g 100 t); holds in vessels, wagons in a train or containers 3.1.6 stratified sampling sampling such that portions of the sample are drawn from the different strata and each stratum is sampled with at least one sampling unit `,,`,-`-`,,`,,`,`,,` - NOTE In some cases, the portions are specified proportions determined in advance However, in post-stratified sampling, the specified proportions would not be known in advance 3.1.7 stratified simple random sampling simple random sampling from each stratum 3.1.8 systematic sampling sampling according to a methodical plan NOTE In bulk sampling, systematic sampling can be achieved by taking items at fixed distances or after time intervals of fixed length Intervals can, for example, be based on mass or time In the case of mass, sampling units or increments should be of equal mass With respect to time, sampling units or increments should be taken from a moving stream or conveyor, for example at uniform time intervals In this case, the mass of each sampling unit or increment should be proportional to the mass flow rate at the instant of taking the entity or increment NOTE If the lot is divided into strata, stratified systematic sampling can be carried out by taking increments at the same relative locations within each stratum 3.1.9 sampling unit 〈bulk material〉 one of the member parts, each with equal probability of selection in sampling, into which a population, comprised of the total quantity of bulk material under consideration, is divided NOTE In bulk sampling, the sampling units are characterized by having an equal probability of being selected Once chosen, the entire sampling unit becomes a part of the whole sample NOTE When sampling from a bulk material is performed by removing individual increments, the sampling unit is the primary increment Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) 3.1.10 precision closeness of agreement between independent test results obtained under stipulated conditions NOTE Precision depends only on the distribution of random errors and does not relate to the true value or the specified value NOTE The measure of precision is usually expressed in terms of imprecision and computed as a standard deviation of test results Less precision is reflected by a larger standard deviation NOTE Quantitative measures of precision depend critically on the stipulated conditions Repeatability and reproducibility conditions are particulate sets of extreme stipulated conditions 3.1.11 bias difference between the expectation of a test result and an accepted reference value NOTE Bias is the total systematic error as contrasted to random error There may be one or more systematic error components contributing to the bias A larger systematic difference from the accepted reference value is reflected by a larger bias value `,,`,-`-`,,`,,`,`,,` - NOTE The bias of a measurement instrument is normally estimated by averaging the error of indication over an appropriate number of repeated measurements The error of indication is the “indication of a measuring instrument less the true value of the corresponding input quantity” 3.1.12 lot 〈bulk material〉 definite part of a population, comprised of the total quantity of bulk material under consideration, and where this part is considered as a quantity of material for which specific characteristics are to be determined NOTE Commerce in bulk material often encompasses transactions involving single lots, and, in these cases, the lot becomes the population 3.1.13 sub-lot 〈bulk material〉 definite part of a lot of bulk material 3.1.14 increment 〈bulk material〉 quantity of bulk material taken in one action by a sampling device NOTE The positioning, delimitation and extraction of the increment should ensure that all parts of the bulk material in the lot have an equal probability of being selected NOTE Sampling is often carried out in progressive mechanical stages, in which case it is necessary to distinguish between a primary increment which is extracted from the lot at the first sampling stage, and a secondary increment which is extracted from the primary increment at the secondary sampling stage, and so on 3.1.15 composite sample 〈bulk material〉 aggregation of two or more increments taken from a lot 3.1.16 gross sample 〈bulk material〉 aggregation of all the increments taken from a sub-lot or lot by the procedures of routine sampling © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) 3.1.17 test sample 〈bulk material〉 sample, as prepared for testing or analysis, the whole quantity or a part of it being used for testing or analysis at one time NOTE The term can be used in such ways as “test sample for chemical analysis”, “test sample for moisture determination”, “test sample for particle size determination” and “test sample for physical testing” 3.1.18 test portion 〈bulk material〉 part of a test sample which is used for analysis or testing at one time 3.1.19 multi-stage sampling 〈bulk material〉 sampling in which the sample is selected by stages, the sampling units at each stage being sampled from the larger sampling units chosen at the previous stage 3.1.20 routine sampling 〈bulk material〉 sampling for commercial purposes carried out by the stipulated procedures in the specific International Standard in order to determine the average quality of the lot NOTE The term “regular sampling” is sometimes used as an alternative to “routine sampling” 3.1.21 experimental sampling 〈bulk material〉 non-routine sampling where special purpose experimental design is applied to investigate sources of variance and/or sampling bias 3.1.22 interpenetrating sampling 〈bulk material〉 replicate sampling from several lots or sub-lots, where for each lot i or sub-lot i, consecutive primary increments are diverted in rotation into different containers to give multiple composite samples (Ai, Bi, Ci,…) in order to investigate the variance between the increments in the lot or the sub-lot NOTE The term “interleaved sampling” is sometimes used as an alternative to “interpenetrating sampling” NOTE Most interpenetrating sampling plans use a duplicate sampling method with composite sample pairs (Ai, Bi) being constituted for each lot i or sub-lot i 3.1.23 replicate sampling 〈bulk material〉 sampling where increments are taken simultaneously or consecutively in pairs, in order to constitute multiple composite samples 3.1.24 duplicate sampling 〈bulk material〉 replicate sampling where increments are taken simultaneously or consecutively in pairs in order to constitute two composite samples NOTE Duplicate sampling is a special case of replicate sampling 3.1.25 manual sampling 〈bulk material〉 collection of increments by human effort 3.1.26 mechanical sampling 〈bulk material〉 collection of increments by mechanical means `,,`,-`-`,,`,,`,`,,` - Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) Put the larger of s e2 ( x ) , s e2 ( y ) in the numerator of Fo and calculate the ratio Fo of the error variances, Fo = s e2 ( x ) /s e2 ( y ) or Fo = s e2 ( y ) /s e2 ( x ) (E.5) If Fo > Fα/2(v1, v2), then the null hypothesis, s e2 ( x ) = s e2 ( y ) , is rejected, and the two groups of data cannot be assumed to be drawn from populations with a common variance The significance level α is usually set equal to 0,05, and v1 and v2 are the degrees of freedom of s e2 ( x ) and s e2 ( y ) respectively, and both are k in this case If Fo < Fα/2(v1, v2), the two groups of data may be assumed to have a common variance 95 % confidence limits, T1(x), T2(x) and T1(y), T2(y) are calculated as follows: T1 ( x ) ,T ( x ) = x ± t 1−α / ( k ) s e ( x ) (E.6) T1 ( y ) ,T ( y ) = y ± t 1−α / ( k ) s e ( y ) (E.7) where x is the grand average of xi1 and xi2; y is the grand average of yi1 and yi2 Calculate: di = s d2 x i1 + x i y i1 + y i − 2 (E.8) k k 1 di − di = k − i =1 k i =1 ∑ ∑ (E.9) A2 = t 1−α / ( f ) / k s d (E.10) `,,`,-`-`,,`,,`,`,,` - If d > A2 , then the hypothesis of the equality of two means is rejected The estimate of bias is d E.5 Residual carbon in heavy oil The quality characteristic is the residual carbon content after 10 % distillation in grade-A heavy oil The reference method is a micro-distillation method specified in the National Standards and it takes about h to obtain a test result The proposed system method under test is a multi-regression equation of readings from the result of spectrophotometry and a test result can be obtained within 10 m Experimental results are shown in Table E.1 Calculations of s e2 ( x ) and s e2 ( y ) are shown in Table E.2 Using Equations (E.2) and (E.4), the following values are obtained: k ∑ g i2 s e2 ( x ) = i =1 2k = 31 = 0,775 40 80 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) s e ( x ) = 0,880 k s e2 ( y ) = ∑ hi2 i =1 2k = 15 830 = 395,750 40 s e ( y ) = 19,893 Putting s e2 ( y ) in the numerator of Fo, Fo = s e2 ( y ) /s e2 ( x ) = 395,75/0,775 = 510,65 Fα / (20,20) = 2,46 < Fo Accordingly, the two groups of data cannot be assumed to be drawn from populations with a common variance 95 % confidence limits, T1(x), T2(x) and T1(y), T2(y) are calculated in accordance with Equations (E.6) and (E.7): T1 ( x ) ,T ( x ) = x ± t 1−α ( k ) s e ( x ) = x ± t 0,975 ( 20 ) s e ( x ) = 342,275 ± 2,086 × 0,880 T1 ( x ) = 340,439, T ( x ) = 344,111 T1 ( y ) , T ( y ) = y ± t 1−α ( k ) s e ( y ) = y ± t 0,975 ( 20 ) s e ( y ) = 312,700 ± 2,086 × 19,893 T1 ( y ) = 271,203, T ( y ) = 354,197 Calculate: x + x i y i1 + y i − d i = i1 2 `,,`,-`-`,,`,,`,`,,` - s d2 k k 1 1 23 486,25 − 591,5 / 20 = 315,402, di − di = = k − i =1 k i =1 19 ∑ A2 = t 1−α / ( ν ) / k ∑ s d = 17,760 s d = 2,093 / 4,472 × 17,76 = 8,312 d = 29,575 > A2 Accordingly, the hypothesis of the equality of two means is rejected The estimate of bias is d = 30 81 © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) Table E.1 — Experimental results of residual carbon in heavy oil Values given as percentage by mass × 103 Sample No System Reference xi1 xi2 yi1 yi2 335 335 318 343 343 342 310 307 340 342 356 300 355 357 372 329 346 347 318 332 348 346 302 320 356 359 325 301 351 353 344 351 343 343 326 317 10 351 351 320 315 11 315 314 296 276 12 329 329 282 283 13 334 334 280 292 14 330 330 353 317 15 331 331 261 312 16 345 345 276 300 17 345 344 280 304 18 333 334 283 317 19 359 359 313 350 20 353 354 299 328 Total 842 849 214 294 342,275 312,700 `,,`,-`-`,,`,,`,`,,` - Average 82 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) Table E.2 — Calculation of s e2 ( x ) and s e2 ( y ) `,,`,-`-`,,`,,`,`,,` - Sample No g i = x i1 − x i g i2 h i = y i1 − y i h i2 0 −25 625 1 −2 56 136 −2 43 849 −1 −14 196 −18 324 −3 24 576 −2 −7 49 0 81 10 0 25 11 1 20 400 12 0 −1 13 0 −12 144 14 0 36 296 15 0 −51 601 −24 576 16 17 −24 576 18 1 −34 156 19 −1 −37 369 20 −29 841 Total 31 15 830 Nonetheless, the error variance of the proposed system is much smaller than the error variance of the reference method It is preferable to try one more experiment, so as to make sure that between duplicate measurements of the proposed system are independent, for example by obtaining duplicate measurements of the proposed system on different days E.6 Filling capacity of tobacco The filling capacity of tobacco is tested in the production of cigarettes in order to assure the hardness of cigarettes Recently, a new filling power tester has been introduced to replace the old one The relationship between the old and the new testers is examined 200 g of test samples are collected from the lot of about 200 kg by the ordinary sampling method Test portions of 15 g for the old tester and of 10 g for the new tester are taken from the test samples Duplicate measurements by the old and the new testers on the lot are shown in Table E.3 83 © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale `,,`,-`-`,,`,,`,`,,` - ISO 11648-1:2003(E) Table E.3 — Filling capacity of tobacco by old and new testers Values in cubic centimetres per gram Lot No Old New xi1 xi2 yi1 yi2 4,590 4,509 4,940 4,969 4,399 4,346 4,792 4,760 4,700 4,700 5,259 5,312 4,562 4,619 5,131 5,114 4,619 4,700 4,846 4,731 4,399 4,481 4,771 4,763 4,183 4,318 4,520 4,516 4,644 4,562 4,910 4,874 5,214 5,189 5,988 6,168 10 5,405 5,270 6,215 6,066 11 5,026 4,998 5,751 5,755 12 4,916 4,916 5,581 5,600 13 5,107 5,026 5,704 5,943 14 4,427 4,456 4,756 4,758 15 4,807 4,782 5,738 5,591 16 4,863 4,888 5,581 5,568 17 5,107 5,132 5,864 5,804 18 4,835 4,888 5,630 5,572 19 5,026 4,970 5,628 5,570 20 5,894 5,866 6,946 6,856 Total 96,723 96,616 108,551 108,290 Average 4,833 5,421 Using Equations (E.2) and (E.4), the following values are obtained: k s e2 ( x ) = ∑ g i2 i =1 2k = 0,086 493 = 0,002 173 40 = 0,171 873 = 0,004 296 40 s e ( x ) = 0,047 k s e2 ( y ) = ∑ hi2 i =1 2k s e ( y ) = 0,066 84 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) Putting s e2 ( y ) in the numerator of Fo, Fo = s e2 (y )/s e2 (x ) = 0,004 296 / 0,002 173 = 1,98 and Fα / (20,20) = 2,46 > Fo The two groups of data may be assumed to have a common variance 95 % confidence limits, T1(x), T2(x) and T1(y), T2(y) are calculated in accordance with Equations (E.6) and (E.7): T1 ( x ) ,T ( x ) = x ± t 1−(α ) ( k ) s e ( x ) = x ± t 0,975 ( 20 ) s e ( x ) = 4,833 ± 2,086 × 0,047 T1 ( x ) = 4,735, T ( x ) = 4,931 T1 ( y ) ,T ( y ) = y ± t 1−(α ) ( k ) s e ( y ) = 5,421 ± 2,086 × 0,066 T1 ( y ) = 5,283, T ( y ) = 5,559 Calculate: di = s d2 x i1 + x i y i1 + y i − 2 k k 2 1 di − di = 8,275 14 − ( −11,751) / 20 = 0,059 117, = k − i =1 k i =1 19 ∑ ∑ s d = 0,243 A2 = t 1−α / ( ν ) / k s d = 2,093 / 4,472 × 0,243 = 0,114 d = 0,588 > A2 The hypothesis of the equality of two means is rejected The estimate of bias is minus 0,588 E.7 COD of exhaust water The chemical oxygen demand (COD) (parameter of water quality) of exhaust water from a plant is regulated by two methods according to the location of the plant One is regulation by total amount of substance relative to the COD, the other is regulation by the mass concentration value of COD, ρCOD, of exhaust water A chemical plant under investigation is located in the area under the regulation of the latter Samples are taken from a storage tank of exhaust water of the chemical plant before dilution by fresh water and CODs are measured Dilution ratios by fresh water are determined based on COD values obtained The standard method specified in National Standards takes about 2,5 h, whereas the rapid method takes 15 to 20 In order to compare the precision and bias between the rapid method and the standard method, CODs are determined in duplicate on the lots by the rapid method and the standard method, respectively Second determinations of duplicate data by the rapid method are obtained on different days, ensuring the independence of the duplicate data Obtained data are shown in Table E.4 `,,`,-`-`,,`,,`,`,,` - 85 © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale `,,`,-`-`,,`,,`,`,,` - ISO 11648-1:2003(E) Table E.4 — COD of exhaust water before dilution Values in milligrams of oxygen per litre Rapid method Sample No 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Average Standard method xi1 xi2 yi1 yi2 850 050 525 575 300 250 875 725 750 201 37 655 788 675 050 180 325 100 650 050 488 340 575 800 400 150 875 413 925 975 025 300 925 400 450 825 215 975 325 229 33 655 650 215 050 200 750 588 913 163 738 345 113 520 400 488 463 388 463 488 248 902 576 004 794 357 699 634 380 240 45 928 001 555 308 764 543 022 815 526 781 517 296 802 895 708 021 570 065 795 512 513 028 491 004 174 656 482 500 229 42 838 987 476 384 740 385 040 823 230 554 554 874 838 745 945 963 555 948 861 395,9 702,7 Using Equations (E.2) and (E.4), the following values are obtained: k s e2 ( x ) = ∑ g i2 i =1 2k = 440 646 = 107 344,1 60 = 468 000 = 24 466,7 60 s e ( x ) = 327,6 k s e2 ( y ) = ∑ hi2 i =1 2k s e ( y ) = 156,4 Putting s e2 ( x ) in the numerator of Fo, Fo = s e2 (x )/s e2 (y ) = 107 344,1/ 24 466,7 = 4,39 and Fα / (30,30) = 2,07 < Fo 86 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale ISO 11648-1:2003(E) Accordingly, the two groups of data cannot be assumed to be drawn from populations with a common variance The 95 % confidence limits, T1(x), T2(x) and T1(y), T2(y), are calculated in accordance with Equations (E.6) and (E.7): T1 ( x ) ,T ( x ) = x ± t 1−α ( k ) s e ( x ) = x ± t 0,975 ( 30 ) s e ( x ) = 395,9 ± 2,042 × 327,6 T1 ( x ) = 726,9, T ( x ) = 064,9 T1 ( y ) ,T ( y ) = y ± t 1−α ( k ) s e ( y ) = y ± t 0,975 ( 30 ) s e ( y ) = 702,7 ± 2,042 × 156,4 T1 ( y ) = 383,3, T ( y ) = 022,1 Calculate: di = s d2 x i1 + x i y i1 + y i − 2 k k 2 1 di − di = 184 108 − ( −9 204 ) / 30 = 81 390,355, = k − i =1 k i =1 29 ∑ ∑ s d = 285,3 A2 = t 1−α / ( v ) / k s d = 2,045 / 5,477 × 285,3 = 106,5 d = 306,8 > A2 `,,`,-`-`,,`,,`,`,,` - Accordingly, the hypothesis of the equality of two means is rejected The estimate of bias is minus 306,8 The urgent task of reduction of the effect of poor precision of the rapid method will be solved by increasing the number of measurements per lot, but a plan for installation of a more precise and somewhat expensive instrument providing an alternative rapid method is being considered E.8 Portable versus installed emission spectrophotometer A portable emission spectrophotometer is introduced for rapid analysis of chemical components in aluminium ingot, through direct emission on the surface of ingot The principle of the portable spectrophotometer is the same as for the installed spectrophotometer At first, a test of the portable spectrophotometer compared with the installed meter is investigated using the method stipulated in ISO 3301 Si and Fe are analysed, but here only Si will be dealt with Results are shown in Table E.5, where xi1 and yi1 are the results on the i-th sample by the portable and installed spectrophotometers, respectively 87 © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) Sample No xi1 yi1 di1 = xi1 − yi1 Di12 1,117 0,895 0,222 0,049 284 3,270 2,418 0,852 0,725 904 0,720 0,620 0,100 0,010 000 2,563 2,026 0,537 0,288 369 1,087 0,922 0,165 0,027 225 2,627 1,898 0,729 0,531 441 0,933 0,783 0,150 0,022 500 1,977 1,518 0,459 0,210 681 0,677 0,627 0,050 0,002 500 10 2,060 1,677 0,383 0,146 689 11 0,453 0,487 −0,034 0,001 156 12 1,613 1,410 0,203 0,041 209 13 1,097 0,965 0,132 0,017 424 14 0,493 0,470 0,023 0,000 529 15 2,187 1,688 0,499 0,249 001 16 0,567 0,482 0,085 0,007 225 17 2,023 1,656 0,367 0,134 689 18 0,457 0,386 0,071 0,005 041 19 1,643 1,491 0,152 0,023 104 20 1,073 0,946 0,127 0,016 129 21 1,120 0,925 0,195 0,038 025 Total 29,757 24,290 5,467 2,548 125 Average 1,417 1,156 0,260 — s d2 21 21 2 1 d i1 − d i1 = 2,548 125 − ( 5,467 ) = 0,056 244 (1) = 20 n − i =1 n i =1 21 to = ∑ d1 s d2 (1) n = ∑ 0,260 0,056 244 21 = 5,030 since t1−α/2(v) = t0,975(20) = 2,086 and the null hypothesis: d = has been rejected Namely, a highly significant bias between the portable spectrophotometer and the installed spectrophotometer has been concluded However, ISO 3301 only makes mention of the comparison of two means and not for testing for bias 88 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale `,,`,-`-`,,`,,`,`,,` - Table E.5 — Determination of Si (%) using portable and installed spectrophotometers ISO 11648-1:2003(E) After a month, an experiment which gives the intermediate precision measured on the same samples a month apart was considered, but the samples used in the above experiment had not been kept Then, a new experiment was designed and two subsequent measurements a day apart on the same sample by the portable spectrophotometer and the installed spectrophotometer were obtained as shown in Table E.6, where xi2 and xi3 were measured using the portable spectrophotometer, and yi2 and yi3 were measured using the installed spectrophotometer Table E.6 — Duplicate measurements of Si (%) using portable and installed spectrophotometers `,,`,-`-`,,`,,`,`,,` - Sample No xi2 xi3 yi2 yi3 1,160 1,085 1,041 1,059 1,525 1,450 1,487 1,455 0,340 0,315 0,342 0,334 1,200 1,100 1,132 1,067 0,455 0,405 0,435 0,428 1,820 1,730 1,657 1,611 0,405 0,355 0,382 0,381 2,030 1,820 1,750 1,717 0,465 0,410 0,435 0,432 10 2,130 1,925 1,807 1,812 11 1,205 1,290 1,126 1,112 12 0,915 0,850 0,909 0,897 13 1,915 1,630 1,633 1,640 14 2,825 2,610 2,410 2,317 15 1,955 1,930 1,680 1,726 16 0,755 0,740 0,724 0,726 17 2,580 2,200 1,682 1,653 18 1,945 1,895 1,743 1,774 19 1,450 1,320 1,275 1,298 20 1,895 1,815 2,081 2,030 21 1,990 1,710 1,630 1,686 Total 30,960 28,585 27,361 27,155 Average 1,417 1,2980 Using Equations (E.2) and (E.4), the following values are obtained: k s e2 ( x ) = ∑ g i2 i =1 2k = 0,512 475 = 0,012 202 42 s e ( x ) = 0,110 89 © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) k s e2 ( y ) = ∑ h i2 i =1 2k = 0,028 152 = 0,000 670 42 s e ( y ) = 0,025 Putting s e2 ( x ) in the numerator of Fo, Fo = s e2 (x )/s e2 (y ) = 0,122 02 / 0,000 670 = 18,21 and Fα / (40,40) = 1,88 < Fo Accordingly, the two groups of data cannot be assumed to be drawn from populations with a common variance The 95 % confidence limits, T1(x), T2(x) and T1(y), T2(y) are calculated in accordance with Equations (E.6) and (E.7): T1 ( x ) ,T ( x ) = x ± t 1−α ( k ) s e ( x ) = x ± t 0,975 ( 21) s e ( x ) = 1,417 ± 2,080 × 0,110 T1 ( x ) = 1,187 7, T ( x ) = 1,647 T1 ( y ) ,T ( y ) = y ± t 1−α ( k ) s e ( y ) = y ± t 0,975 ( 21) s e ( y ) = 1,298 ± 2,080 × 0,025 T1 ( y ) = 1,244 1, T ( y ) = 1,351 Calculate: x + x i3 y i + y i3 − d i = i2 2 s d2 k k 1 di − di = 0,965 752 75 − 2,514 / 21 = 0,033 23, = k − i =1 k i =1 20 ∑ ∑ s d = 0,182 d = 0,120 > A2 Accordingly, the hypothesis of the equality of two means is rejected The estimate of bias is d = 0,120 Since confidence intervals T1(x), T2(x) and T1(y), T2(y) overlap, it is inconclusive whether or not a bias exists between the portable spectrophotometer and the installed spectrophotometer Every effort, such as more careful grinding of the surface of ingot to be emitted, is being devoted to the improvement of the precision of the portable spectrophotometer 90 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale `,,`,-`-`,,`,,`,`,,` - A2 = t 1−α / ( ν ) k s d = 208 / 4,472 × 0,182 = 0,085 ISO 11648-1:2003(E) Bibliography [1] COCHRAN, W.G The relative accuracy of systematic and stratified random samples from a certain class of populations Annals of Mathematical Statistics, 17, 1946, p 164 [2] COCHRAN, W.G Sampling Techniques John Wiley, New York, 1953 [3] AOKI, Shigeo Sample division for size determination Reports of statistical application research, JUSE, 29 (4), 1982 [4] AOKI, Shigeo and YONEDA, T Sample division of bulk materials Reports of statistical application research, JUSE, 34 (4), 1987 [5] SATTERTHWAITE, F.E Biometrics Bull 2, 1946 [6] ANDERSON, R.L and BANCROFT, T.A Statistical theory in research, McGraw-Hill Book Co., Inc., New York, 1952 [7] Confidence limit of variance component, in JUSE Statistical Table B, 1982 ed [8] JOWETT, G.H The accuracy of systematic sampling from conveyor belts Applied Statistics, 1952 [9] HEBDEN, J and JOWETT, G.H The accuracy of sampling coal, Applied Statistics, 1952 [10] JOWETT, G.H and SCOTT, J.F Simple graphical techniques for calculating serial and spatial correlations, and mean semi-squared differences J.R Statist Soc., B, 15, 1953, p 81 [11] JOWETT, G.H The comparison of means of industrial time series Applied Statistics, 1953 [12] W ARD, D.H Weekly, monthly, and quarterly tolerances for coke quality Applied Statistics, 1959 [13] GY, P.M Sampling of Particulate Materials — Theory and Practice Elsevier Scientific Publishing Co., Amsterdam, 2nd ed., 1982 [14] GY, P.M Sampling of heterogeneous and dynamic material systems — Theories of heterogeneity, sampling and homogenizing Elsevier Scientific Publishing Co., Amsterdam, 1992 [15] GY, P.M L’échantillonnage des lots de matière en vue de leur analyse Masson publ., 1996 [16] ISO 3084, Iron ores — Experimental methods for evaluation of quality variation [17] ISO 3085, Iron ores — Experimental methods for checking the precision of sampling, sample preparation and measurement [18] ISO 3086, Iron ores — Experimental methods for checking the bias of sampling [19] ISO 3301, Statistical interpretation of data — Comparison of two means in the case of paired observations [20] SNEDECOR, G.W and COCHRAN, W.G Statistical Methods IOWA State University Press, 7th ed., 1980 [21] AOKI, Shigeo Case studies for bias testing Bulletin for sampling research JUSE, 237, February 1991 [22] GRUBBS, F.E On estimating precision of measuring instruments and product variability J.A.S.A., 1948 [23] THOMSON, G.J JR Precision of simultaneous measurement procedures, J.A.S.A., 1963 `,,`,-`-`,,`,,`,`,,` - 91 © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) HAHN, G.J and NELSON, W.A Problem in the statistical comparison of measuring devices Technometrics, 1970 [25] MALONEY, C.J and RASTOGI, S.C Significance test for Grubb’s estimators Biometrics, 1970 [26] JAECH, John L Further tests of significance for Grubb’s estimators Biometrics, December 1971 [27] GRUBBS, F.E Errors of measurement, precision, accuracy and the statistical comparison of measuring instruments Technometrics, 1973 [28] ISO 2854, Statistical interpretation of data — Techniques of estimation and tests relating to means and variances [29] ISO 8258, Shewhart control charts [30] ISO 10725, Acceptance sampling plans and procedures for the inspection of bulk materials [31] Guide to the expression of uncertainty in measurement (GUM), BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, 1st ed., 19931) `,,`,-`-`,,`,,`,`,,` - [24] 1) Corrected and reprinted in 1995 92 Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS © ISO 2003 — All rights reserved Not for Resale `,,`,-`-`,,`,,`,`,,` - Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS Not for Resale ISO 11648-1:2003(E) ICS 03.120.30 Price based on 92 pages © ISO 2003 — All rights reserved Copyright International Organization for Standardization Provided by IHS under license with ISO No reproduction or networking permitted without license from IHS `,,`,-`-`,,`,,`,`,,` - Not for Resale